Astrophysics Cosmic Magnetic Field From Jcost Primordial Bcert

PrimordialBCert, and where it stops

A ledger, a discrete record of events, sits at the center of Recognition Science. The declaration PrimordialBCert is a small, machine-checked package of three facts about the framework's recognition cost, the forced price of recognizing one event in terms of another. It bundles three general truths about the cost function J(x) = (x + 1/x)/2 - 1: the cost is zero when the two quantities are equal, the cost never goes negative for positive inputs, and the golden-ratio threshold phi - 3/2 is positive. Each fact is proved in the framework's machine-checked library of formal theorems, and the certificate simply groups them together.

The name PrimordialBCert suggests a statement about the early universe's magnetic field, but the certificate itself never mentions magnetism, cosmology, or any physical quantity. Its cost, the price of recognition, is defined as J(m / e) for two real numbers m and e, with no definition of what m and e mean. The research note attached to the module records an intended application: a primordial magnetic field strength B_RS = J(phi) times the equipartition value, roughly 0.118 times 10^(-8) gauss, which saturates the Planck observational upper bound of 10^(-9) gauss. That note is a plan, not a result. The formal theorems prove only the three general facts about J, universally quantified over real inputs.

What the certificate does establish is genuine, if modest. It shows that the cost function vanishes at equality, that it is nonnegative on positive inputs, and that the golden-ratio threshold is positive. These are the building blocks for any later application, but they are not themselves an astrophysical claim. The certificate's own documentation says it proves nothing specific to its subject, because the domain cost is defined without reference to one. To make PrimordialBCert a theorem about cosmic magnetic fields, one would need a definition of m and e in that subject's own terms, say a mass and a charge, and a derivation that the ratio m / e actually controls the field strength. That step remains open.

In Recognition Science, the framework models the cosmological magnetic field upper bound as a saturation of the observed limit, but this is a research note, not a proved theorem. The honest takeaway: PrimordialBCert is a valid, machine-checked certificate of three general properties of the cost function, and nothing more. Its name is an aspiration, not a proof. A reader who wants the cosmic magnetic field result must wait for a definition that ties m and e to physical quantities, a step the framework has not yet taken.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

PrimordialBCert does not prove any bound on the cosmic magnetic field. The certificate does not define m and e as mass and charge. The research note's saturation of the Planck bound is a plan, not a proved result.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/CosmicMagneticField_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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